A hybrid Eulerian–Lagrangian approach for non-self-similar expansion analysis of a cylindrical cavity in saturated and unsaturated critical state soils

This paper proposes a powerful hybrid Eulerian–Lagrangian (HEL) approach for the analysis of cavity expansion problems. The new approach is applied to analysing the non-self-similar expansion process of a hollow cylinder of critical state soils, considering arbitrary saturation states of soils and both drained and undrained conditions. A closed-form solution for the stresses and displacements in the elastic zone is presented, taking the state-dependent soil moduli and outer boundary effect of the soil cylinder into account. Adopting large strain theory in the plastic zone, the non-self-similar cavity expansion process is formulated into a set of partial differential equations in terms of both Eulerian and Lagrangian descriptions, which is solved by a newly proposed algorithm. The HEL approach is compared with the conventional Eulerian and Lagrangian approaches for the cavity expansion analyses. It is found that the new approach can reduce to the Eulerian approach when the self-similar assumption is satisfied and to the Lagrangian approach when stress–total strain relationships are obtained analytically. Finally, the expansion process is proven to be non-self-similar by showing the stress and deformation paths, and the finite thickness of soil cylinders may greatly influence the cavity expansion behaviour, especially with a small thickness ratio. The HEL approach can provide useful tools for validating advanced numerical techniques for both saturated and unsaturated soils and interpreting pressuremeter tests in small-size calibration chambers.